The modulator’s frequency is both added to and subtracted from the carrier’s frequency; the resulting harmonics replace the original carrier and modulator. The result is a more complex set of harmonics that don’t follow typical “musical” spacing based on octaves above the fundamental; as such, it is useful for creating clangorous, metallic “inharmonic” sounds. Want an example? Say the carrier was a sinewave (only the fundamental harmonic present) at 600Hz, and the modulator was a sine wave at 100Hz. The result would be a tone that had frequency components at 500 and 700Hz.

Why do you need a special ring modulator module instead of just a VCA? Because VCAs usually don’t have a concept of negative loudness, cutting off half of the result and yielding a different harmonic structure where the carrier is still present.